Jaroslav Hron

Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow

We describe new benchmark settings for the rigorous evaluation of different methods for fluid-structure interaction problems. The configurations consist of laminar incompressible channel flow around an elastic object which results in self-induced oscillations of the structure. Moreover, characteristic flow quantities and corresponding plots are provided for a quantitative comparison.

A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics

We investigate a new method of solving the problem of fluid-structure interaction of an incompressible elastic object in laminar incompressible viscous flow. Our proposed method is based on a fully implicit, monolithic formulation of the problem in the arbitrary Lagrangian-Eulerian framework. High order FEM is used to obtain the discrete approximation of the problem. In order to solve the resulting systems a quasi-Newton method is applied with the linearized systems being approximated by the divided differences approach. The linear problems of saddle-point type are solved by a standard geometric multigrid with local multilevel pressure Schur complement smoothers. 1 Overview We consider the problem of viscous fluid flow interacting with an elastic body which is being deformed by the fluid action. Such a problem is encountered in many real life applications of great importance. Typical examples of this type of problem are the areas of aero-elasticity, biomechanics or material processing. For example, a good mathematical model for biological tissue could be used in such areas as early recognition or prediction of heart muscle failure, advanced design of new treatments and operative procedures, and the understanding of atherosclerosis and associated problems. Other possible applications include the development of virtual reality programs for training new surgeons or designing new operative procedures (see [12]). 1.1 Fluid structure models in biomechanics There have been several different approaches to the problem of fluid-structure interaction. Most notably these include the work of [13, 14, 15, 16] where an immersed boundary method was developed and applied to a three-dimensional model of the heart. In this model they consider a set of one-dimensional elastic fibers immersed in a three-dimensional fluid region and using a parallel supercomputer they were able to model the pulse of the heart ventricle. ⋆ This work has been supported by German Reasearch Association (DFG), Reasearch unit 493. A fluid-structure model with the wall modelled as a thin shell was used to model the left heart ventricle in [3, 4] and [18, 17]. In [7, 8] a similar approach was used to model the flow in a collapsible tube. In these models the wall is modelled by two-dimensional thin shells which can be modified to capture the anisotropy of the muscle. In reality the thickness of the wall can be significant and very important. For example in arteries the wall thickness can be up to 30% of the diameter and its local thickening can lead to the creation of an aneurysm. In the case of heart ventricle the thickness of the wall is also significant and also the direction of the muscle fibers changes through the wall. 1.2 Theoretical results The theoretical investigation of fluid structure interaction problems is complicated by the need of mixed description. While for the solid part the natural view is the material (Lagrangian) description, for the fluid it is the spatial (Eulerian) description. In the case of their combination some kind of mixed description (usually referred to as the arbitrary Lagrangian-Eulerian description or ALE) has to be used which brings additional nonlinearity into the resulting equations. In [10] a time dependent, linearized model of interaction between a viscous fluid and an elastic shell in small displacement approximation and its discretization is analyzed. The problem is further simplified by neglecting all changes in the geometry configuration. Under these simplifications by using energy estimates they are able to show that the proposed formulation is well posed and a global weak solution exists. Further they show that an independent discretization by standard mixed finite elements for the fluid and by nonconforming discrete Kirchhoff triangle finite elements for the shell together with backward or central difference approximation of the time derivatives converges to the solution of the continuous problem. In [19] a steady problem of equilibrium of an elastic fixed obstacle surrounded by a viscous fluid is studied. Existence of an equilibrium state is shown with the displacement and velocity in C and pressure in C under assumption of small data in C and domain boundaries of class C. A numerical solution of the resulting equations of the fluid structure interaction problem poses a great challenge since it includes the features of nonlinear elasticity, fluid mechanics and their coupling. The easiest solution strategy, mostly used in the available software packages, is to decouple the problem into the fluid part and solid part, for each of those parts to use some well established method of solution then the interaction is introduced as external boundary conditions in each of the subproblems. This has an advantage that there are many well tested finite element based numerical methods for separate problems of fluid flow and elastic deformation, on the other hand the treatment of the interface and the interaction is problematic. The approach presented here treats the problem as a single continuum with the coupling automatically taken care of as internal interface, which in our formulation does not require any special treatment. 2 Continuum description Let Ω ⊂ R be a reference configuration of a given body. Let Ωt ⊂ R 3 be a configuration of this body at time t. Then a one-to-one, sufficiently smooth mapping χΩ of the reference configuration Ω to the current configuration χΩ : Ω × [0, T ] 7→ R , (1) describes the motion of the body, see figure 1. The mapping χΩ depends on the choice of the reference configuration Ω which can be fixed in a various ways. Here we think of Ω to be the initial (stress-free) configuration Ω0. Thus, if not emphasized, we mean by χ exactly χΩ = χΩ0 .

Simple flows of fluids with pressure–dependent viscosities

In his seminal paper on fluid motion, Stokes developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure. Such an assumption is particularly well suited to modelling flows of many fluids at high pressures and is relevant to several flow situations involving lubricants. Fluid models in which the viscosity depends on the pressure have not received the attention that is due to them, and we consider unidirectional and two–dimensional flows of such fluids here. We note that solutions can have markedly different characteristics than the corresponding solutions for the classical Navier–Stokes fluid. It is shown that unidirectional flows corresponding to Couette or Poiseuille flow are possible only for special forms of the viscosity. Furthermore, we show that interesting non–unique solutions are possible for flow between moving plates, which has no counterpart in the classical Navier–Stokes theory. We also study, numerically, two two–dimensional flows that are technologically significant: that between rotating, coaxial, eccentric cylinders and a flow across a slot. The solutions are found to provide interesting departures from those for the classical Navier–Stokes fluid.

Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities

There is a considerable amount of experimental evidence that unequivocally shows that there are fluids whose viscosity depends on both the mean normal stress (pressure) and the shear rate. Recently, global existence of solutions for the flow of such fluids for the three-dimensional case was established by Malek, Necas and Rajagopal. Here, we present a proof for the global existence of solutions for such fluids for the two-dimensional case. After establishing the global-in-time existence, we discretize the equations via the finite element method, outline the Newton type iterative method to solve the non-linear algebraic equations and provide numerical computations of the steady flow of such fluids in geometries that have technological significance.

Numerical Benchmarking of Fluid-Structure Interaction: A Comparison of Different Discretization and Solution Approaches

Comparative benchmark results for different solution methods for fluid-structure interaction problems are given which have been developed as collaborative project in the DFG Research Unit 493. The configuration consists of a laminar incompressible channel flow around an elastic object. Based on this benchmark configuration the numerical behavior of different approaches is analyzed exemplarily. The methods considered range from decoupled approaches which combine Lattice Boltzmann methods with hp-FEM techniques, up to strongly coupled and even fully monolithic approaches which treat the fluid and structure simultaneously.

Numerical Simulation and Benchmarking of a Monolithic Multigrid Solver for Fluid-Structure Interaction Problems with Application to Hemodynamics

An Arbitrary Lagrangian-Eulerian (ALE) formulation is applied in a fully coupled monolithic way, considering the fluid-structure interaction (FSI) problem as one continuum. The mathematical description and the numerical schemes are designed in such a way that general constitutive relations (which are realistic for biomechanics applications) for the fluid as well as for the structural part can be easily incorporated. We utilize the LBB-stable finite element pairs Q 2 P 1 and P 2 + P 1 for discretization in space to gain high accuracy and perform as time-stepping the 2nd order Crank-Nicholson, respectively, a new modified Fractional-Step-θ-scheme for both solid and fluid parts. The resulting discretized nonlinear algebraic system is solved by a Newton method which approximates the Jacobian matrices by a divided differences approach, and the resulting linear systems are solved by direct or iterative solvers, preferably of Krylov-multigrid type.

Flows of Incompressible Fluids subject to Navier's slip on the boundary

We establish closed form analytical solutions for the flows of a generalized fluid of complexity two, which includes the Navier-Stokes fluid, power-law fluid and the second-grade fluid as special subclasses, in special geometries under the assumption that the flows meet Navier slip conditions at the boundary. The boundary conditions allow the extremes of no-slipping and no-sticking. We also allow for different boundary conditions at different parts of the boundary, which leads to interesting consequences with regard to the solutions. The dependence of the form of the solutions on the boundary conditions is discussed in some detail.

A numerical investigation of flows of shear‐thinning fluids with applications to blood rheology

A new solver is presented for the flow of power-law fluids that extends a solver developed by Turek [FEATFLOW] for the Navier-Stokes fluid. This solver is convenient for simulating efficiently both steady and unsteady flows of shear-dependent fluids in a complex geometry. To illustrate the ability of the solver, two specific problems are chosen. First, steady flows of power-law fluids are studied in corrugated channels, and qualitative comparisons with real experiments are carried out. Attention is paid to the dependence of friction factor and dimensionless normal stress amplitude on the aspect ratio (amplitude versus wavelength of the sinusoidal channel) and to the occurrence of secondary flows. It is shown that the aspect ratio is not a sensible non-dimensional number in this geometry. Second, unsteady (pulsatile) flows of the power-law fluid (i.e. blood under certain circumstances) are simulated in the presence of stenosis and a very good coincidence with recent numerical studies is obtained. The description of the numerical scheme and theoretical background are also outlined

A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes

We present special numerical simulation methods for non-isothermal incompressible viscous fluids which are based on LBB-stable FEM discretization techniques together with monolithic multigrid solvers. For time discretization, we apply the fully implicit Crank-Nicolson scheme of 2nd order accuracy while we utilize the high order Q 2 P 1 finite element pair for discretization in space which can be applied on general meshes together with local grid refinement strategies including hanging nodes. To treat the nonlinearities in each time step as well as for direct steady approaches, the resulting discrete systems are solved via a Newton method based on divided differences to calculate explicitly the Jacobian matrices. In each nonlinear step, the coupled linear subproblems are solved simultaneously for all quantities by means of a monolithic multigrid method with local multilevel pressure Schur complement smoothers of Vanka type. For validation and evaluation of the presented methodology, we perform the MIT benchmark 2001 M.A. Christon, P.M. Gresho, S.B. Sutton, Computational predictability of natural convection flows in enclosures, in: First MIT Conference on Computational Fluid and Solid Mechanics, vol. 40, Elsevier, 2001, pp. 1465-1468] of natural convection flow in enclosures to compare our results with respect to accuracy and efficiency. Additionally, we simulate problems with temperature and shear dependent viscosity and analyze the effect of an additional dissipation term inside the energy equation. Moreover, we discuss how these FEM-multigrid techniques can be extended to monolithic approaches for viscoelastic flow problems.

On the Modeling of the Synovial Fluid

Synovial fluid is a polymeric liquid which generally behaves as a viscoelastic fluid due to the presence of hyaluronan molecules. We restrict ourselves to the regime in which the fluid responds as a viscous fluid. A novel generalized power-law fluid model is developed wherein the power-law exponent depends on the concentration of the hyaluronan. Such a model will be adequate to describe the flows of such fluids as long as they are not subjected to instantaneous stimuli. Assuming two different structures for the form of the power law exponent, both in keeping with physical expectations, we numerically solve for the flow of the synovial fluid (described by the constraint of incompressibility, the balance of linear momentum, and a convection-diffusion equation for the concentration of hyaluronan) in a rectangular cavity. The solutions obtained with our models are compared with the predictions of those based on a model that has been used in the past to describe synovial fluids. While all the three models seem to agree well with available experimental results, one of the models proposed by us seems to fit the data the best; it would, however, be hasty to pass judgment based on this one particular experimental correlation.